Let $\mathsf{I}$ and $\mathsf{J}$ be equivalent categories. Let $\mathsf{C}$ be another category. I need to prove that the categories of functors $[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$ are equivalent.
Let functors $E\colon\mathsf{I}\to\mathsf{J}$, $E'\colon\mathsf{J}\to\mathsf{I}$ together with natural isomorphisms $\eta\colon 1_{\mathsf{I}} \Rightarrow E'E, \epsilon\colon EE' \Rightarrow 1_{\mathsf{J}}$ define an equivalence of categories.
Define a functor $y\colon[\mathsf{I},\mathsf{C}]\to[\mathsf{J},\mathsf{C}]$ which maps each functor $F\colon\mathsf{I}\to\mathsf{C}$ to $FE'$ and each natural transformation $\alpha\colon F\Rightarrow G$ to the natural transformation $\beta\colon FE'\Rightarrow GE'$ such that for any $j \in \mathsf{J}$ we have $\beta_j = \alpha_{E'(j)}$.
I have proven that this functor is faithful and essentially surjective on objects, but I'm not sure how to prove that it is full.