Functors map diagrams to diagrams and cones to cones. We are naturally tempted to ask if the image of a limiting cone is also a limiting cone in the target category. We first study some very small examples in which the answers to this question are either yes or no depending on the functor in question. Then, we will study the \({{\,\textrm{Hom}\,}}\) functors that guarantee the positive answers for all limiting cones.