Cofiber sequence of a wedge summand inclusion

The inclusion U→U∨V is the pushout of the basepoint inclusion {* }→V along {* }→U. That inclusion is an unbased cofibration by well-pointedness, so F2 proves the claim. Collapsing U identifies the remaining V only at its existing basepoint, and compatible quotient maps in both directions give $(U\vee V)\mathrm{/}U\cong V$. F4 consequently identifies its cofiber with V up to based homotopy.

More explicitly that cofiber is $CU\vee V$: the original U is the base of the attached cone. On CU use $[u,s]\mapsto [u,s+t-st]$ and keep V fixed. At t=0 this is the identity, at t=1 all of CU is the tip, and the basepoint track is fixed. The quotient-times-I construction in F2 makes this a continuous based deformation onto V. The cofiber projection to ΣU collapses all of V, so its composite with the inclusion V→CU∨V is identically the basepoint. Under this explicit inverse of the equivalence, the connecting map is therefore constant. For U=V=S1 this gives the stated two-circle instance.