$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$

If it exists, a weak inverse is unique up to natural isomorphism, and furthermore can be improved to form an adjoint equivalence, where $\iota$ and $\epsilon$ sastisfy the triangle identities.

More generally, given a $2$-category$\mathcal{B}$ and a morphisms $F: C \to D$ in $\mathcal{B}$, a weak inverse of $F$ is a morphism $G: D \to C$ with $2$-isomorphisms

$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$

Given the geometric realization of categories functor $\vert -\vert: Cat \to Top$, weak inverses are sent to homotopy inverses. This is because the product with the interval groupoid is sent to the product with the topological interval $[0,1]$. In fact, less is needed for this to be true, because the classifying space of the interval category is also the topological interval. If we define a lax inverse to be given by the same data as a weak inverse, but with $\iota$ and $\epsilon$ replaced by natural transformations, then the classifying space functor sends lax inverses to homotopy inverses. An example of a lax inverse is an adjunction, but not all lax inverses arise this way, as we do not require the triangle identities to hold.

(David Roberts: I’m just throwing this up here quickly, it probably needs better layout or even its own page.)