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There is a map $[A, B] \times [B, A] \to [A, A] \times [B, B]$ given by composition in each of the possible directions. There is also a map $1 \to [A, A] \times [B, B]$ which picks out $(\text{id}_A, \text{id}_B)$. These fit together into a pullback diagram $[A, B] \times [B, A] \to [A, A] \times [B, B] \leftarrow 1$ and the object of isomorphisms is the limit (so pullback) of this diagram.