I’ve been trying to prove the following claim, but am now unsure about its truth. Is it true, and if so, where can I find a proof?

**Claim:** For any categories C, D, E such that C and D are equivalent,

(i) The set $Hom($C, E$)$ of functors from C to E is in bijective correspondence with the set $Hom($D, E$)$ of functors from D to E, i.e. $Hom($C, E$)$ $simeq$ $Hom($D, E$)$.

(ii) The set $Hom($E, C$)$ of functors from E to C is in bijective correspondence with the set $Hom($E, D$)$ of functors from E to D, i.e. $Hom($E, C$)$ $simeq$ $Hom($E, D$)$.