Sketch

Remarks

The sketch is a limit sketch when $C$ is empty. Some authors use a directed graph with chosen diagrams instead of the category $\mathcal{A}$; this gives the same models.

Example: let $\mathcal{A}$ be the free category on the graph with two objects $X, P$ and three arrows $p_1, p_2, m \colon P \to X$. Let $L$ contain the single cone $(P; p_1, p_2)$ over the discrete diagram $(X, X)$, and let $C$ be empty. A model in sets is a functor $M$ with $M(P) \cong M(X) \times M(X)$ through $M(p_1), M(p_2)$. Then $M(m)$ is a binary operation on $M(X)$, so models are sets with a binary operation.

Non-example: let $M(X) = M(P) = \set{0, 1}$ with $M(p_1) = M(p_2) = M(m) = \id$. This functor is not a model, since $\set{0, 1}$ with two identity maps is not a product of $\set{0, 1}$ with itself. Indeed, the pair of maps $(\id, \text{constant } 0)$ does not factor through it.