I've been reading about the complement system, as part of the human immune system. The complement system is introduced as part of the article on innate immunity on Wikipedia. This classification makes obvious sense in the case of the alternative pathway, which is obviously not dependent on any specific pathogen.
However, I'm confused about whether this is an appropriate classification for the classical pathway (and, therefore, the system writ large) since the classical pathway is activated by antigen:antibody complexes and only by certain antibody isotypes (IgG and IgM). It is, therefore, to some extent pathogen specific and it is also, clearly, activated by and dependent upon the humoral immune response (generally classed as part of adaptive immunity).
So, in short, is it best to classify the complement system as part of the innate immunity (and, if so, why is that?), or would it be better to restrict ourselves to the observation that it is, to a greater or lesser extent, part of both types of immunity?