The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
$\textbf{Theorem}$ Let $D$ be a diagram in $\textbf{K}$ with sets $V-$indexed family and every $E-$indexed family of objects in $\textbf{K}$ has a product and if every pair of $\textbf{K-}$morphisms (between the same two objects) has an equalizer, $D$ has a limit.
Proof: Form the $V-$indexed and $E-$indexed products shown below. The universal property of the $E-$indexed product induces unique $\psi_1$ and $\psi_2$ as shown.
Let $h=eq(\psi_1, \psi_2)$. Set $\gamma_i=\pi_i\circ h$ as shown above. We claim that $(L,(\gamma_i))$ is a cone over $D$. If $e:i\rightarrow j$ is in $E$, we must show
But
$$D_e\circ \gamma_i=D_e\circ pi_i\circ h=\bar{\pi}_e\circ\psi_2 \circ h=\bar{\pi}_e\circ \psi_1\circ h=\pi_j\circ h=\gamma_j.$$
Now suppose that $(L',(\gamma^{'}_i))$ is an arbitrary cone over $D$. By the universal property of the $V-$indexed product, there exists unique $h'$ with $pi_i\circ h'=\gamma^{'}_i.$ (Look at $\textbf{(4)}$ imagining primes where necessary.) Consulting $\textbf{(5)}$, substituting primes where necessary, we have
$$\bar{\pi}_e\circ\psi_1 \circ h'=\pi_j\circ h'=\gamma^{'}_j=D_e\circ \gamma^{'}_i=D_e\circ \pi_i\circ h'=\bar{\pi}_e\circ\psi_2 \circ h'.$$
As $e$ is arbitrary, $\psi_1 \circ h'=\psi_2 \circ h'$. Since $h=eq(\psi_1, \psi_2)$ there exists unique $\Gamma:L'\rightarrow L$ such that $h\circ \Gamma=h'.$
Thus
$$h_i\circ\Gamma=\pi_i\circ h\circ \Gamma=\pi_i\circ h'=h^{'}_i.$$
If also $\bar{\Gamma}$ satisfies $h_i\circ \bar{\Gamma}=h^{'}_i$ then as $\pi_i\circ h\circ \bar{\Gamma}=\pi_i\circ h^{'}$ for all $i,$ $\bar{\Gamma}\circ h=h^{'}$ and so $\Gamma=\bar{\Gamma}.$
I just have a quick $\textbf{question}$ about the above proof:
(1) what do the notation: $(D_j\mid i\xrightarrow{e} j \in E)$ mean?
(2) In the proof, where it says "Look at $\textbf{(4)}$ imagining primes where necessary.) Consulting $\textbf{(5)}$, substituting primes where necessary..." what did it mean by imagining primes where necessary", and then to ask the reader to consult $\textbf{(5)}$ for the purpose of substituting primes....? What did the authors mean by that statement, I mean imagining primes and when is it necessary to make the necessary substitution?
Thank you in advance
I was confused at first. They don't mean "primes" as in prime numbers. They mean "primes" as in the "prime" notation, the "dash": they say, look at diagram $(4)$ but e.g. replace $h$ with $h'$. The "where necessary" is to indicate that, in this changed situation where $(L',\gamma')$ is some other cone, replace all the related arrows with their dashed form: $\gamma_i$ to be replaced with $\gamma'_i$, etc.
The same goes for the other comment. They just mean, by replacing all arrows in $(5)$ with their dashed forms ('primes') we know $D_e\circ\gamma'_i=\gamma'_j$.
Oh, and about the notation $(D_j:i\overset{e}{\longrightarrow}j\,\text{ in $E$})$. They are taking the following product: if $A$ is the set of all arrows of $E$, that is, the set: $\{e:i\overset{e}{\longrightarrow}j\,\text{ in $E$}\}$, then we are considering: $$\prod_{e\in A}D_j$$Where for a given coordinate $e$, "$j$" is the codomain of $e$. It is convenient to express this as: $$\prod_{i\to j}D_j$$