Understanding the definition of "preserving limits"

Question

From Mac Lane's Category Theory:

A functor $H:C \rightarrow D$ is said to preserve the limits of functors $F: J \rightarrow C$ when every limiting cone $v: b \dot \rightarrow F$ in $C$ for a functor $F$ yeilds by composition with $H$ a limiting cone $Hv : Hb \dot \rightarrow Hf$ in $D$. A functor that is continuous when it preserves all small limits.

Is this saying: For some collection of functors $F:J \rightarrow C$ from some $J$ into $C$ and their unique limiting cones $v : b \dot \rightarrow F$, it must be the case that $Hv : Hb \dot \rightarrow HF$ for each of these functors is a limiting cone?

And do these $Hv$ functors have to be the same TYPE of limiting cone as the functor $v$? As in products to products, equalizers to equalizers etc...

Answer 1

What is being defined here is the concept of "preserving $J$-shaped limits". I'm not sure exactly what your question is, but let me state the definition with all the quantifiers explicitly mentioned.

Let $H:C\to D$ be a functor and $J$ be a category. Then we say that "$H$ preserves $J$-shaped limits" (or "$H$ preserves the limits of functors $J\to C$") if:

For every functor $F:J\to C$ and every limiting cone $v:b\dot\to F$, $Hv:Hb\dot\to HF$ is also a limiting cone.

As for your last question, $HF$ is a functor $J\to D$. So there is only one possible "type" of limiting cone that $Hv$ could be, namely the limiting cone of a $J$-shaped limit. So yes, $Hv$ is always the same type of limiting cone as $v$ was.