If Caitlyn has Tank, or if there are multiple units with Tank including Caitlyn, how do the damage assignment rules work?

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If Caitlyn, Patrolling is given Tank, Tank’s rules govern assignment: all units with Tank controlled by the same player must be assigned lethal damage before any non-Tank units [815.1.c.2]. If Caitlyn, Patrolling has a printed requirement to be assigned last as well as Tank, the assigning player must choose which of those mutually-exclusive assignment requirements to follow for that unit [460.2.c.7]. If multiple units have such conflicting requirements, each unit is decided individually; if that results in multiple units sharing the same priority, they may be assigned in any order within that priority [460.2.c.8]. Always obey the general requirements to assign lethal damage fully before moving on and not to overassign beyond lethal unless no other units remain [460.2.c.3] [460.2.c.4].

Tank forces assignment to Tank units before any non-Tank units for that controller; if Caitlyn, Patrolling also has a printed “assign last” requirement, the assigning player chooses whether to treat Caitlyn, Patrolling as Tank (assign first) or as printed (assign last) — they cannot partially satisfy both [815.1.c.2] [460.2.c.7]. If multiple units have the same mandatory priority after choices are made, assign them in any order among themselves [460.2.c.6] [460.2.c.8].

Identify which of your units have Tank. Tank units must be assigned lethal damage before any non-Tank units controlled by the same player [815.1.c.2]. If more than one Tank exists for that controller, you may assign damage to those Tank units in any order among themselves [460.2.c.6]. Remember that you must assign lethal damage to a unit in full before moving to another unit and you may not assign more than the minimum lethal amount unless no further units remain to receive damage [460.2.c.3] [460.2.c.4]. If a unit (for example Caitlyn, Patrolling) has Tank and also a printed instruction requiring it be assigned last, the assigning player chooses which of those mutually-exclusive assignment requirements to apply for that unit; you may not partially satisfy both [460.2.c.7]. If multiple units each have conflicting requirements (e.g., multiple Caitlyns that also have Tank), decide for each unit individually which requirement to apply; if those choices create multiple units sharing the same priority, assign them in any order within that priority [460.2.c.8]. If any unit cannot be dealt damage, treat it as exempt from mandatory assignment considerations (it is not counted toward lethal-assignment ordering) [460.2.c.9].

If Caitlyn, Patrolling has Tank and also a printed "assign last" instruction, you cannot assign her damage in the middle of other units because that would satisfy neither requirement — you must choose to apply either Tank (assign her before non-Tank units) or her printed "last" effect (assign her after other applicable units) [460.2.c.7]. If two or more units each have both Tank and printed ordering effects, handle each unit’s choice separately — it's possible to choose to apply Tank on some copies and the printed text on others; after choices, if multiple units share the same assignment priority, you may order them freely among themselves [460.2.c.8] [460.2.c.6]. A unit that cannot be dealt damage is ignored for these assignment priorities and does not count as a valid target for lethal-assignment ordering [460.2.c.9].

Tank is a passive ability that functionally reads “I must be assigned lethal damage before any other unit with the same controller as me that does not have [Tank] during the Combat Damage step” [815.1.b]. Practically, when assigning combat damage you first consider which of your opponent’s units have Tank; you must assign lethal damage to all of those Tank units before assigning damage to any of their non-Tank units [815.1.c.2]. Additionally, general combat damage assignment rules require you to assign lethal damage to a unit in full before moving on and not to assign more than the minimum lethal amount unless no other units remain to receive damage [460.2.c.3] [460.2.c.4].

Where things conflict (for example, Caitlyn, Patrolling is printed to be assigned last but has been given Tank, which says assign first), the rules treat that as mutually-exclusive assignment requirements on the same unit. The assigning player must choose which single requirement to apply for that unit when assigning damage; they may choose to treat Caitlyn, Patrolling as Tank (assign her before non-Tank units) or to honor her printed "assign last" text (assign her after other applicable units). They cannot split the assignment (e.g., assign her in the middle) because that would satisfy neither requirement [460.2.c.7].

If multiple units each have such conflicting requirements (multiple Caitlyns each given Tank), handle each unit separately — choose which requirement applies for each unit, then resolve assignments. If, after making those choices, multiple units share the same required priority (for example, two units now both must be assigned before non-Tank units because you chose Tank for both), those units can be assigned in any order among themselves [460.2.c.8] [460.2.c.6].

Finally, if a unit cannot be dealt damage at all (due to some effect), it is exempt from being considered for lethal-assignment ordering — no amount of damage is considered lethal to it, so you do not have to assign lethal to it before others [460.2.c.9].

Summary of application to your question: If Caitlyn, Patrolling is given Tank, she becomes a Tank unit and must be assigned lethal damage before non-Tank units for that controller unless you instead choose to apply some other mutually-exclusive printed requirement (like "assign last") to her; make that choice per unit, then follow the normal rules about assigning lethal in full and ordering among units that share the same priority [815.1.c.2] [460.2.c.7] [460.2.c.3].