Please, check if my proof is correct.
Let $\mathcal{H}$ be the category with Objects the topological spaces, Morphisms the homotopy classes of continous maps. In this category, isomorphisms are homotopy equivalences.
To show: If $X$ is path connected and $X$ is isomorphic to $Y$, then $Y$ is path connected.
Proof:
The path components of $X$ are in bijection with $\mathcal{H}(*,X)$ (the homotopy classes of maps from the one point set to X). If $X$ is path connected there is exactly one such map. Since $X$ and $Y$ are isomorphic in $\mathcal{H}$, there is a bijection between $\mathcal{H}(*,X)$ and $\mathcal{H}(*,Y)$. This means, that $\mathcal{H}(*,Y)$ also has exactly one element.
This proof is correct. The only suggestion I have is to perhaps emphasize that your first fact implies that a space is path-connected if and only if $\mathcal{H}(\ast,X)$ is a one-point set (you only mention one direction of this equivalence explicitly) since the proof uses both directions.