We present dynamic Monte Carlo lattice simulations of the coil to globule collapse of single chains of a copolymer comprising monomer units, m and c, wherein there is a net attractive interaction between c-units. As the copolymer is cooled, the solvent quality becomes poorer, and the size of the chain decreases, driven by the net m-m and c-c attractions. The strong c-c attraction increases the overall solvophobicity of the chain relative to a homopolymer and, therefore, copolymers collapse more abruptly and at a higher effective temperature relative to homopolymers. We compare copolymers with homopolymers by rescaling collapse data to the same theta values to account for the effect of overall solvophobicity. This comparison shows that the behavior of copolymers and the corresponding homopolymers is identical as the chain size reduces from high temperatures to the theta value. Beyond theta, copolymers with c-content < similar to 50% collapse more abruptly than either homopolymer, after accounting for the difference in overall solvophobicity. Collapse of copolymers containing higher c-content is dominated entirely by the c-c attractions, and these chains behave qualitatively like homopolymers with a higher effective solvophobicity. Analysis of the chain structure during collapse provides a structural reason for the qualitative change in copolymer collapse at low c-content. When such copolymers are cooled below theta, the c-units rapidly aggregate to form an isotropic, compact core surrounded by an anisotropic solvated shell of m-units. The shell densifies as the copolymer is further cooled, but remains anisotropic for the finite chain sizes investigated.