TITLE: Universal property of the cocomplete category of models of a limit sketch
QUESTION [5 upvotes]: Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\mathcal{E} \to \mathbf{Set}$ which send the cones in $\mathscr{S}$ to limit cones) is cocomplete, in fact locally presentable. It enjoys the following universal property in the $2$-category of cocomplete categories:  If $\mathcal{C}$ is any cocomplete category, then
$$\mathrm{Hom}_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C}) \simeq \mathbf{Mod}_{\mathcal{C}}(\mathscr{S}^{\mathrm{op}}).$$
Here, $\mathrm{Hom}_c$ denotes the category of cocontinuous functors, $\mathscr{S}^{\mathrm{op}}$ denotes the dual colimit sketch, and $\mathbf{Mod}_{\mathcal{C}}$ refers to $\mathcal{C}$-valued models.
A typical example of this universal property is $\mathrm{Hom}_c(\mathbf{Grp},\mathcal{C}) \simeq \mathbf{CoGrp}(\mathcal{C})$, that is, $\mathbf{Grp}$ is the universal example of a cocomplete category with an internal cogroup object.
The proof is a combination of two well-known results, so I assume that this universal property is also well-known. Can someone confirm this and point me to literature which I can cite for this result?
Maybe it follows from some of the theorems in Kelly's book on enriched categories, chapter 6. But I am not sure. 

Here is a sketch of the two-step proof I was thinking of. Let $G : \mathbf{Mod}(\mathscr{S}) \to [\mathcal{E},\mathbf{Set}] $ be the inclusion and $F : [\mathcal{E},\mathbf{Set}] \to \mathbf{Mod}(\mathscr{S})$ its left adjoint. Then by a "tensor-less" variant of Prop. 2.3.6. in Tensor functors between categories of quasi-coherent sheaves (as you see, this is not quite the reference I need!) the category $\mathrm{Hom}_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C})$ is equivalent to the category of those cocontinuous functors $Q^* : [\mathcal{E},\mathbf{Set}] \to \mathcal{C}$ such that $Q^* \to Q^* \circ G \circ F$ is an isomorphism; this is true iff the right adjoint $Q_* : \mathcal{C} \to [\mathcal{E},\mathbf{Set}]$ factors over $\mathbf{Mod}(\mathscr{S})$. By the well-known universal property of $[\mathcal{E},\mathbf{Set}]$ as the free cocompletion of $\mathcal{E}^{\mathrm{op}}$ (for instance, Section 4.4 in Kelly's book), $Q^*$ corresponds to a functor $P : \mathcal{E}^{\mathrm{op}} \to \mathcal{C}$ via $Q^* = P \otimes_{\mathcal{E}} -$, and hence $Q_* = \mathrm{Hom}(P(-),-)$. By the Yoneda Lemma, the requirement that $Q_*$ factors over $\mathbf{Mod}(\mathscr{S})$ exactly means that $P$ is a model of $ \mathscr{S}^{\mathrm{op}}$.

I have found a reference: Theorem 2.2.4 in this paper (actually, I knew this paper, but I somehow forgot that the theorem is in there). But the authors also explicitly state that this theorem is well-known. I would be happy with a more "classical" or "original" reference.

REPLY [2 votes]: Probably the earliest reference is Theorem 2.5 in

A. Pultr, The right adjoints into the categories of relational systems, Reports of the Midwest Category
Seminar IV. Springer, Berlin, Heidelberg, 1970

Pultr's "relational theories" are exactly small realized limit sketches.
Remark: I have recently generalized the result to large limit sketches (arXiv:2106.11115).