# Left adjoint for the “strings category” functor

Let $\mathbf{Cat}$ be the category of small categories and let $\mathbf{sCat}$ denote the category of simplicial objects in $\mathbf{Cat}$. We have a functor $$\text{str}\colon \mathbf{Cat}\longrightarrow \mathbf{sCat}$$ taking a category $\mathcal{C}$ to the simplicial object whose category of $n$-simplices is given by $\mathcal{C}^{[n]}$ and whose faces and degeneracies are given by precomposition with the cofaces and the codegeneracies in the simplex category.

Question: is there a left adjoint for this functor? If so, how can it be described explicitly?

EDIT: by looking at the simplex category as a discrete 2-category, this may actually be an instance of the enriched version of the fact that presheaf categories are the free cocompletion of ordinary (small) categories. Still, I'd like to have an explicit description of the left adjoint, which I can't immediately see using enriched Kan extensions.

Let $I : \Delta \to \mathrm{Cat}$ be the "inclusion" that sends $[n]$ to the category also called $[n]$ that you mentioned in your description of $\mathrm{str}$. Then $\mathrm{str}(\mathcal{C}) = \mathrm{Fun}(I(-), \mathcal{C})$. The left adjoint is then given by $- \otimes I$, the enriched functor tensor product with $I$, computed by an enriched coend $\mathcal{X} \mapsto \int^{[n] \in \Delta} \mathcal{X}_n \times [n]$. (Here, $\mathcal{X}$ is a simplicial object in categories, $\mathcal{X}_n$ is its category of $n$-simplices, and the $[n]$ that $\mathcal{X}_n$ is multiplied by is really the category $I([n])$.)