Let $M$ be a model of an algebraic theory $\mathcal T$. $M$ is said to be finitely generated if it is the quotient of a free model $F(n)$ over a finite set $n$. Here, quotient means there exists a regular epi $F(n)\twoheadrightarrow M$, (i.e this epi is a cokernel).
From the adjunction $F\dashv U$, this arrow corresponds to an arrow $n\rightarrow U(M)$, which is just an $n$-tuple $(u_1,\dots,u_n)$. Hence, it seems to me that the condition $M$ be a quotient of a free model over a finite set can be replaced simply by "there is an arrow to $M$ from a free model over a finite set".
What am I missing? Why should $M$ indeed be a quotient? I thought finitely generated was exactly about specifying arrows = specifying images of the basis.