What is the defintion of "H preserves all small products and equalizers"?

Question

If $C$ is a complete category, and $H:C \rightarrow D$ preserves all small products and all equalizers (of parallel pairs) prove that $H$ is continuous.

I'm trying to prove this theorem but I'm not sure what "preserves all small products and all equalizers" means.

Does it mean: $C$ has ALL products and ALL equalizers and if $\bar c$ is a product in $C$ and $f,g$ are parallel arrows in $C$ with equalizer $e$ that $H\bar c$ is a product in $D$ and $Hf, Hg$ are parallel arrows with equalizer $He$?

Or does it mean: for the products and equalizers THAT EXIST in $C$, ALL $H\bar c$ and $Hf, Hg$ and $He$ as previously defined are products and equalizers in $D$?

Answer 1

From user k.stm' comments:

It means the latter.

However, as "$C$ is a complete category" it also means the former in this case.