I have a *really tame* category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is non-empty, then $C(y,x)$ is empty. I want to fatten this category a little without changing the homotopy type of the classifying space.

Let $\mu$ be a subset of the morphisms in $\bigsqcup_{x,y \in C_0}C(x,y)$. For each $f:x \to y$ in $\mu$, I want to add $g:y \to x$ subject to the relations $fgf = f$ and $gfg = g$. Call this new $g$-enhanced category $C[\mu]$. First question,

Is $C \to C[\mu]$ a standard operation and does it have a name? Clearly, this is similar to $-$ but not quite the same as $-$ localization around $\mu$.

Assume now that our selection of morphisms $\mu$ has been chosen so that there are no zig-zag cycles in $C$ of the form $$x_0 \to y_0 \leftarrow x_1 \to y_1 \to \ldots \leftarrow x_k \to y_k \leftarrow x_0,$$ where all the $x_j$ and $y_j$ are distinct, and only the forward maps lie in $\mu$. Thus, the addition of the corresponding $g$-morphisms does not destroy any existing loops in the classifying space $BC$. Second, and more important question,

Does the zig-zag acyclicity of $\mu$ suffice to guarantee homotopy equivalence $BC[\mu] \simeq BC$ of classifying spaces? If not, what more do we need from $\mu$ to achieve this homotopy equivalence?