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For a diagram shape $I$, a functor $F:\mathcal{C}\to \mathcal{D}$ preserves categorical limits of shape $I$ if every limiting cone $(L,{\ell }_i)$ over every diagram $D:I\to \mathcal{C}$ is carried to a limiting cone $(FL,F{\ell }_i)$ over $FD$. It reflects categorical limits of that shape if a cone in $\mathcal{C}$ is limiting whenever its image cone is limiting. These properties concern cones which are given; reflection is not an assertion that every cone in the target lifts.

Suppose $F$ is a full and faithful functor and the image of $(L,{\ell }_i)$ is limiting. A cone $(X,x_i)$ in $\mathcal{C}$ becomes a cone in $\mathcal{D}$, so there is a unique $h:FX\to FL$ with $(F{\ell }_i)h=F{x}_i$. Fullness writes $h=Fk$ for some $k:X\to L$. Faithfulness gives ${\ell }_{i}k=x_i$, and also ensures uniqueness: two such $k$ would have the same image, by uniqueness of $h$. Hence the original cone is limiting. This proves every full and faithful functor reflects categorical limits, the limits reflected by full and faithful functors criterion, with no essential-surjectivity requirement.