I'm pretty sure this has an easy solution, but I can't seem to find it.
Let $X$ be a contractible $2$-dimensional CW-complex, let $\gamma$ be an embedded loop in $X$, and let $f : D^2 \rightarrow X$ be an embedding of a disc in $X$ which maps the boundary of $D$ to $\gamma$.
My question is the following. Let $f' : D^2 \rightarrow X$ be a continuous map of a disc into $X$ which takes the boundary of $D$ to $\gamma$. Must we then have $f(D^2) \subset f'(D^2)$ ? I'm pretty sure that the answer is yes, but I can't seem to prove it.
Of course, this has an obvious generalization to higher dimensional complexes, and I'd be interested in that too.