Why is the full subcategory consisting of simply connected spaces not complete? Let $\mathbf{Top}_*$ be the category of pointed topological spaces and $\mathbf{Top_1}$ the full subcategory of simply connected spaces. $\mathbf{Top}_*$ is complete and cocomplete. I am trying to show that $\mathbf{Top_1}$ is not complete.
If $X$ is simply connected but has a non-trivial $\pi_2$-group, then $\Omega X$ (the based loop space) is not simply connected. Also, $\Omega X$ is the pullback (in $\mathbf{Top}_*$) of the path space fibration $PX \to X$ along the map $* \to X$. For example, one could take $X = S^2$.
This shows that the pullback, if it exists in $\mathbf{Top_1}$, is not the same as the pullback computed in $\mathbf{Top}_*$. 
Can I use this to show that the pullback doens't exist in $\mathbf{Top_1}$? Can I somehow show that if it did exist, then "it must be $\Omega X$"? Perhaps one can consider the relationship with the underlying sets, i.e. $\mathbf{Set}$? 

Suppose a diagram of simply connected pointed spaces has limit $X$ in $\mathbf{Top}_*$. If it has limit $\tilde{X}$ in $\mathbf{Top}_1$ then there is a canonical map $\tilde{X}\to X$ that is the universal map from a simply connected pointed space to $X$.
The universal cover of $X$ (when it exists) comes close to this, but the failure of lifting theorems for spaces that are not locally path connected stops it working.
As noted in Jack Davies' comment, $X=S^1$ is the pullback of a pair of maps of simply-connected spaces. If a map $\tilde{X}\to X$ existed with the required universal property, then by considering paths (maps from $[0,1]$) and homotopies between them (maps from $[0,1]\times[0,1]$) it's fairly easy to see that it would have to be the universal cover $\mathbb{R}\to S^1$. However, there are maps $Y\to S^1$ from simply connected but not locally path connected spaces $Y$ that do not lift to maps $Y\to\mathbb{R}$.
