How to understand the "create limit"? I find it is hard to understand the "create limit." (You can find it in Mac Lane's Categories for the working mathematician, P112; there it defines: "A functor $V:A→x$ creates limits for a functor $F:J→A$.")
Here are my questions:
(1) Why does it use "a functor $F:J→A$," and how does it match with the common limit in the algebra (before category)?
2) Does the "creates limits" always exist? If not, please give me some examples.
3)How does the forgetful  functor create limits? Take the the forgetful functor $U:Grp→Set$, for example; what is "the functor $F:J→A$"?
 A: IMO you are wondering, how limits in analysis relate to limits in category theory. AFAIK they are not related — simple. Instead, limits in category theory are a generalization of the greatest lower bound in a preorder.
At elementary level limits are defined for “diagrams” (you should know what are they). But every functor can be interpreted as a diagram. So people just talk about limits for functors. The functor $F:J\to A$ is interpreted as the diagram in the category $A$. $J$ is the category called a “scheme” of the diagram $F$. E.g. the scheme of binary product diagrams is the free category over the graph consisting of 2 vertexes and no edges.
It is important that functors map not only objects and morphisms, but also diagrams. This is obvious, because a diagram is a bunch of objects and morphisms. $V:A\to X$ maps $F$ to $V(F)$. $F, V(F)$ have the same scheme. Morphisms in $F$ commute, $F$ preserves compositions of morphisms, then morphisms in $V(F)$ commute, then $V(F)$ is a diagram. If you look at $F$ as a functor, then $V(F)$ is actually $V\circ F$.
“$V:A\to X$ creates limits” means that for every diagram $F$ in $A$ and every limit $\tau$ of $V(F)$ there exists a unique limit of $F$ which is a preimage (along $V$) of $\tau$. E.g. if we choose a particular limit $F$ — the binary product diagram of some groups $G_0, G_1$, we choose a particular functor $V$ — the forgetful functor $Grp\to Set$, and $V(G_0)\times V(G_1)$ — the Cartesian product of the carriers (underlying sets) of those groups — is the limit of $V(F)$, then there exists a unique group with the carrier $V(G_0)\times V(G_1)$ (it has this carrier because it is the preimage of the set $V(G_0)\times V(G_1)$), and and that group is the categorical product of $G_0, G_1$ with projections $\pi_0:V(G_0)\times V(G_1)\to V(G_0), \pi_1:V(G_0)\times V(G_1)\to V(G_1)$ understood as group homomorphisms. I.e. that a Cartesian product of carriers can be extended to a categorical product of groups, which is called also a direct product of groups.
