The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
$\text{Definition 1:}$ A $\textbf{product}$ in the category $\textbf{K}$ of a a family of objects $(A_i\mid i\in I)$ is an object $A$ together with a family
$$(A \xrightarrow{\normalsize{\text{in}}_i} A_i\mid i\in I)$$ of morphisms (called $\textbf{projections}$) with the property that given any other object $C$ similarly equipped with an $\textit{I}-$indexed family $(f_i:C\to A_i)$ there exists a
unique morphism $f:C\to A$ such that ${\pi}_i\circ f=f_i$ for all $i$ in $I.$ We write $A=\prod_{i\in I}A_i,$ If $I=\{1,\ldots,n\}$ we may write $A=A_1\times\ldots\times A_n.$
$\textbf{Proposition:}$ Given a family $(A_i\mid i\in I$ of vector spaces, we define their $\textbf{product}$ to be the cartesian product $$\prod_{i\in I}=\{f\mid f:\to \cup_{i\in I} A_i\text{ and } f(i)\in A_i \text{ for each }i\}$$
equipped with the 'coordinatewise' addition and multiplication-by-a-scalar $$(f+f")(i)=f(i)+f'(i)\quad \text{ and } (\lambda \cdot f)(i)(i)=\lambda\cdot f(i)$$
for all $f,f'$ in $\prod_{i\in I}A_i,\lambda$ in $\textbf{R}$ and $i\in I.$ Then $\prod_{i\in I}A_i$ together with the projections $$\pi_j:\prod_{i\in I}A_i\to A_i:f\mapsto f(j)$$
is a product of $A_i)$ in the category $\textbf{Vect}$ in the sense of $\text{Definition 1:}$
Proof: Just check that $p(c)(i)=p_i(c)$ does the trick in $\text{Definition 1:}.$ Don't forget to check that $\pi_j$ and $pa$ are linear!
Question, I would like to know in the proof of the above proposition, in the notation $p(c)(i)=p_i(c),$ what does $p(c)(i)$ mean? Does it mean $p(c_i)$?
Thank you in advance
$p$ in the sense of definition (1) is the unique map $f : C \to A$.
Note that your starting data is the collection $p_i$ (that is, $f_i : C \to A_i$). Also, we are working inside the concrete category of vector spaces.
Now, define $p$ (that is, $f : C \to \prod_i A_i$ by, $f(c) : i\mapsto f_i(c)$). You just need to check that this map $p$ is the unique arrow $C \to \prod_i A_i$ that makes all the small triangles (given in your question) commute. Once you do this, you have basically shown that the given choice of $(A, (\pi_i))$ does satisfy the Universal Property of Products in a category, and so, $(A, (\pi_i))$ is the product (categorical product) of the family $A_i$.