Enriching categories and equivalences Let $\mathcal{C}$ and $\mathcal{D}$ be  two equivalent categories. Furthermore, assume  $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to enrich $\mathcal{D}$  over $(\mathcal{M}, \otimes)$? 
 A: I think most category theorists would answer "yes, obviously", and not bother to write down a proof.  But presumably that isn't sufficiently convincing, since you ask the question, so let me try to make it a bit more explicit with some big words.  (-:
An $M$-enriched category with set of objects $A$ is equivalently a lax functor $A_{ch}\to B M$, where $A_{ch}$ is the chaotic category with $A$ as objects (exactly one morphism between any two objects) and $B M$ is the 1-object bicategory associated to $M$.  A function $f:A\to B$ induces a functor $f_{ch}: A_{ch} \to B_{ch}$, hence by precomposition a map $f^*$ from $M$-categories with object set $B$ to $M$-categories with object set $A$, such that $f^*\underline{B}(a_1,a_2) = \underline{B}(f(a_1),f(a_2))$ for any $M$-category $\underline{B}$ with object set $B$.
Now suppose that $C$ is an $M$-category with object set $B$, that $D$ is an ordinary category with object set $A$, and $f:D \to C_0$ is a fully faithful and essentially surjective functor, where $C_0$ is the underlying ordinary category of $C$.  Then $f$ induces a function $A\to B$ and thereby an induced $M$-category $f^*C$ with object set $A$.  Moreover, the action of $f$ on homs consists of bijections $D(x,y) \cong C_0(f(x),f(y))$ that respect composition and identities; but by construction of $f^*C$ these are equivalently $D(x,y) \cong (f^*C)_0(x,y)$, giving an isomorphism of categories $D\cong (f^*C)_0$, i.e. an enrichment of $D$ over $M$.
Note that nowhere did we use the essential surjectivity of $f$, so actually we've proven a bit more.
