I know a Sledgehammer is a special type of hammer, but I still do not quite get the exact meaning of the word in such a paragraphs as:
The computational sledgehammer par excellence is the spectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.
Does it mean something like the key to decipher a problem, to facilitate or easy up computation, or does it mean something like the nemesis in this context, a fatal flaw which makes everything crash?