In CW-homology, we define the chain complex of a CW-complex as $C_n=H_n(Sk_n X,Sk_{n-1} X)$ with boundary maps given by $d$ in the following diagram (denoting $X_{n}=Sk_n X$): 
The diagram allows us to see that $H_n^{cell} (X):= ker(d)/im(d)=H_n(Sk_n X)/im(\delta_n)=H_n(Sk_{n+1} X)=H_n^{sing} (X)$.
In practice I see people intepreting the reasonably complicated boundary map $d$ conretely: for example in the case of $\mathbb{RP}^2=S^2/(x\sim -x)$, which has precisely one $n$-cell for $n=0,1,2$, we attach $D^1$ via the double cover, yielding a disk with doubled boundary $2u$. Now a single $2$-cell $D^2$ is attached via the double cover $\alpha: S^1\rightarrow Sk_{1} X$, then one interprets the boundary map $d: C_{2}\rightarrow C_{1}$ as $d:D^2\mapsto 2u$.
Question: What is the theoretical basis for interpreting the CW-boundary map $d$ as the intuitive topological boundary of the attached $n$-cells in general? In my above example I can draw some commutative diagrams and make it work, but (is there) what is the rigour in general?