Let $Z$ be $\Omega Y$ which is the space of loops based at $Y_0$. Then I know how to define a map explicitly from $Z \to \Omega \Sigma X$. It is defined by noting we have the identity map $ \Sigma Z, \Sigma Z$ and then using the adjoint relation $[\Sigma Z, \Sigma Z] \to [Z \to \Omega \Sigma Z]$. Explicitly this is given by sending $z$ to the inclusion $[z \times [-1,1] \hookrightarrow \Sigma Z$.
In this paper http://www.math.rochester.edu/people/faculty/jnei/mooreexp.pdf on page 8, he says there is a map $\Omega$-eval a map the other way around. How do you define it?
You can't say that this is $\gamma(0)$ for a loop $\gamma$ in the loop space of $\Sigma Z$ because you can't be sure that it will lie in $Z$. What else could you do?
From $Id\in [\Omega Y,\Omega Y]$ and using the usual adjunction, you get a map $ev\in [\Sigma\Omega Y, Y]$. His map $\Omega \Sigma \Omega Y \to \Omega Y$ is just $\Omega(ev)$.
(Note that $ev(\lambda\wedge t)=\lambda(t)$, this explains the terminology).