A functor $F : A \to B$ is called consistent if it satisfies Frei’s “C condition” (Frei, 1976), which following (Hilton, 1966) p. 241 is equivalent to the following condition: for all $A\in A_o$, $FA \cong \varprojlim \big( FA\downarrow F \to A \xto{F} B \big)$, and which, following Linton (Linton, 1969) is equivalent to the fact that $\exp_F^\op$ is fully faithful on the pairs $(B,FA)$.
Bibliography
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Frei, A. (1976). On categorical shape theory. Cahiers De Topologie Et Géométrie Différentielle Catégoriques, 17(3), 261–294.Details
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Hilton, P. (1966). Correspondences and Exact Squares. In S. Eilenberg, D. K. Harrison, S. MacLane, & H. Röhrl (Eds.), Proceedings of the Conference on Categorical Algebra (pp. 254–271). Springer Berlin Heidelberg.Details
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Linton, F. E. J. (1969). An outline of functorial semantics. In B. Eckmann (Ed.), Seminar on Triples and Categorical Homology Theory (pp. 7–52). Springer Berlin Heidelberg.Details