What is the standard terminology regarding the arrows and objects involved in pullbacks? In Awodey's Category Theory (2006) he discusses the pullback of arrows $f:A \rightarrow C$, $g: B \rightarrow C$ (in a category $\mathbf{C}$) as consisting of arrows $p_1: P \rightarrow A$, $p_2: P \rightarrow B$ such that $f \circ p_1 = g \circ p_2$ and this property is universal (p. 92). So it seems that a pullback has three parts, the object $P$ (which Awodey doesn't put a lot of emphasis on), and the two arrows $p_1,p_2$.
On nLab's page on pullbacks, they refer to the object $P$ as the pullback, which makes the arrows $p_1$ and $p_2$ seem almost incidental, whereas for Awodey they seemed central. (I know the arrows aren't merely "incidental", the point is a matter of emphasis.) But then I don't know what to call the arrows involved ($p_1$ and $p_2$). 
Given a setup like the one above, what are some standard ways of talking about $P$, $p_1$, and $p_2$, especially expressions that indicate their relationship to the objects $A$, $B$, and $C$ and arrows $f$ and $g$?   
 A: Let $I$ be a small category; in the case of pullbacks, take $I = (\bullet \rightarrow \bullet \leftarrow \bullet)$. Let $X : I \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$ of shape $I$. Then a limit of $X$ is a final object in the category of cones to $X$ . Thus, it consists of a universal cone, sometimes called the limit cone (or limiting cone) $(p_i : L \to X(i))_{i \in I}$  such that for every cone $(q_i : T \to X(i))_{i \in I}$ there is a unique morphism $f : T \to L$ such that $p_i f = q_i$. It follows that every two limit cones are uniquely isomorphic (in the category of cones, of course), so that one may speak of the limit cone*. The morphisms $p_i : L \to X_i$ are called projections.
The object $L$, considered separately, has no significance, and it does not really make sense to call it a limit of $X$. Quite often mathematicians are just lazy and say "Let $L$ be a limit of $X$" ("with projections $p_i : L \to X(i)$") when they actually mean "Let $(p_i : L \to X(i))_{i \in I}$ be a limit cone" (which is, actually, shorter!). Nevertheless, Mac Lane calls $L$ a limit object and uses, as many others, the notation $L = \varprojlim X$. So in the case of pullbacks, this would be pullback object.
Also notice that $L$ is not unique up to unique isomorphism, so that it is not really unique in a categorical sense. In order to formulate unicity, we  really have to remember the projections.
*It is very common to identify objects which are uniquely isomorphic. For example, one speaks of the trivial group, although (in ZFC) for every set $X$ we can endow the set $\{X\}$ with the trivial group structure $X \cdot X = X$. Thus there are as many trivial groups as sets. But they are all isomorphic in a unique way, so that there is no harm to identify them. More generally, one can identify all final objects of a category; and a limit cone of $X$ is just a final object in the category of cones to $X$.
