Equivalence and isomorphism of $\hom$s in $2$-categories/bicategories

Question

In a usual $(1)$-category $C$, if $A$ is an object and $B\cong B'$ are isomorphic objects, then the hom-sets $\hom(A,B)\cong \hom(A,B')$ are in bijective correspondence, because composition with the isomorphism $B\cong B'$ yields a bijection.

Now, consider a bicategory or $2$-categories $C$, i.e. the $\hom$s are not sets but categories. The objects of such a $\hom(A,B)$ are then called 1-morphisms and the morphisms of such a $\hom(A,B)$ $2$-morphisms. There are now two notions of sameness between objects:

• $A$ and $B$ are isomorphic if there are 1-morphisms $f\colon A\to B$ and $g\colon B\to A$ such that $gf=id$ and $fg=id$.

• $A$ and $B$ are equivalent if there are 1-morphisms $f\colon A\to B$ and $g\colon B\to A$ such that $gf$ and $fg$ are isomorphic to $id$ (in the respective $\hom$-categories), as witnessed by pairs of 2-morphisms.

Now, fix objects $A$, $B$, and $B'$. Questions:

1. Let $B$ and $B'$ be isomorphic. Does it follow that the category $\hom(A,B)$ is isomorphic to $\hom(A,B')$? Does it follow that the category $\hom(A,B)$ is equivalent to $\hom(A,B')$?

2. Let $B$ and $B'$ be equivalent. Does it follow that the category $\hom(A,B)$ is equivalent to $\hom(A,B')$? Does it follow that the category $\hom(A,B)$ is isomorphic to $\hom(A,B')$?

Answer 1

As Zhen Lin said in the comments:

In a (strict) 2-category isomorphic objects have isomorphic hom-categories, and equivalent objects have equivalent hom-categories.