Recall that a locally small category $\mathcal{C}$ is called total if the Yoneda embedding$Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a left adjoint $L$.

Definition

A locally small category $\mathcal{C}$ is called lex total if the Yoneda embedding$Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a left exact (=finite limit preserving) left adjoint $L$.

The result was announced by Walters on the Isle of Thorns in 1976, a proof can be found in Street (1981, p.206). More detailed information on this characterization in particular concerning the size issues involved and the algebraic perspective it avails can be found at Grothendieck topos or the blog posts by Bob Walters.