Let $${\mathcal {M}}$$ M be an arbitrary collection of linear-fractional non-automorphism self-maps of the unit disk $$\mathbb {D}$$ D . We consider the unital $$\hbox {C}^*$$ C∗ -algebras $$C^*(T_z, \{C_{\varphi } : \varphi \in {\mathcal {M}}\})$$ C∗(Tz,{Cφ:φ∈M}) and $$C^*(\{C_{\varphi } : \varphi \in {\mathcal {M}}\}, {\mathcal {K}})$$ C∗({Cφ:φ∈M},K) generated by the composition operators induced by the maps in $${\mathcal {M}}$$ M and either the unilateral shift $$T_z$$ Tz or the ideal of compact operators $${\mathcal {K}}$$ K on the Hardy space $$H^2(\mathbb {D})$$ H2(D) . We describe the structures of these $$\hbox {C}^*$$ C∗ -algebras, modulo the ideal of compact operators, for all finite collections $${\mathcal {M}}$$ M as well as all collections $${\mathcal {M}}$$ M that have finite boundary behavior. This work completes a line of research investigating the structures, modulo the ideal of compact operators, of Toeplitz-composition and composition $$\hbox {C}^*$$ C∗ -algebras induced by linear-fractional non-automorphism self-maps of $$\mathbb {D}$$ D that has unfolded over the last decade and half. While all results in this paper are stated for $$H^2(\mathbb {D})$$ H2(D) , the descriptions of the structures of these $$\hbox {C}^*$$ C∗ -algebras, modulo the ideal of compact operators, also apply to the weighted Bergman spaces $$A^2_{\alpha }(\mathbb {D})$$ Aα2(D) for $$\alpha > -1$$ α>-1 .