From Mac Lane's Category Theory:
In the red box below, why are the components of $\tau$ written as $\tau_x$? Since $\tau$ is a cone from $X$ to $F$, it should be $\tau : \Delta X \rightarrow F$ where $\Delta X$ is the constant functor sending each $j \in J$ to $X$. Therefore each component of $\tau$ should be $\tau_j : X \rightarrow F_j$.
How does the author get $\tau_x$ as the components and why does what is in the red box complete the proof?
On
Maybe it's better to see the limit in more elementary terms first. The limit $L$, if it exists, is a pair $(L\in Set,(\nu_j)_{j\in J})$ that makes the following diagram commute:
\begin{array}{cc} \,\,\,\,\,\,\,L \\ \\ \ \nu_j \downarrow & \,\,\,\,\,\,\,\searrow \,{\nu_i} \\ \\ Fj & \xleftarrow{F(i\overset{f}{\rightarrow}j)} & F_i \end{array}
and that satisfies the usual universal mapping property: if $(X\in Set,(\tau_j)_{j\in J})$ satisfies $Ff\circ \tau_i=\tau_j$ then there is a unique $\phi:X\to L$ such that $\nu_j\circ \phi=\tau_j.$
It's not too hard to show that $L$ is the set of choice functions $\left \{ \psi:J\to F_j \right \},\ $ i.e. the set of $j$-tuples $L=\left \{ (x_j)_{j\in J} \right \},$ such that $x_j\in Fj$ and $Ff\circ \pi_i=\pi_k.\ $ (so $\nu_i=\pi_i,\ $ the $i$-th projection).
Now, any particular $j$-tuple $x=(x_j)_{j\in J}$ in $L$ can be realized as a collection of functions $\left \{ \nu_j(x) :*\to F_j \right \}_{j\in J},$ where $\nu_j(x)$ sends the singleton $*$ to the $j$-th component of $( x_j)_{j\in J} .\ $
And since $Ff\circ \nu_i(x)(*)=Ff(x_i)=Ff\circ \pi_i((x_j)_j)=\pi_j(x_j)_j=x_j=\nu_j(x)(*),\ $ each $\left \{ \nu_j(x) :*\to F_j \right \}_{j\in J}$ is a cone from $*$ to $F.$
Thus, $L$ is the same thing as the set of cones $\left \{ \nu_j(x):*\to F_j \right \}_{j\in J}$ as $x$ ranges over $L.$
Now, if $(X\in Set,(\tau_j)_{j\in J})$ is as above, then for each $x\in X,\ \tau(x)=(\tau_j(x))_{j\in J}$ so it is $\textit also$ a cone from the singleton; that is, $\left \{ \tau_j(x):*\to F_j \right \}_{j\in J}$
Finally, for the unique function $\phi:X\to L,$ it suffices to take $\phi(x)=\left \{ \nu_j(\tau(x)) :*\to F_j \right \}_{j\in J}. $
$\tau x$ is not a component of the cone $\tau$; instead, for every $x\in X$ the collection of elements $(\tau_j(x))_j$ is a cone to $F$ from the singleton, which is denoted $\tau x$. So by definition $\tau x\in Cone(*,F)$, and this defines a function $h:X\to Cone(*,F)$. Now the commutativity can be checked for all $x\in X$; indeed, if $h(x)=(\tau_j(x))_j$, then $v_j(h(x))=\tau_j(x)$ by the definition of the map $v_j$. It follows that $v_j\circ h=\tau_j$ for all $j$. The uniqueness follows from the definition of the $v_j$'s.