  We study nonintersecting Brownian motions with two prescribed starting and
ending positions, in the neighborhood of a tacnode in the time-space plane.
Several expressions have been obtained in the literature for the critical
correlation kernel $K\tac(x,y)$ that describes the microscopic behavior of the
Brownian motions near the tacnode. One approach, due to Kuijlaars, Zhang and
the author, expresses the kernel (in the single time case) in terms of a
$4\times 4$ matrix valued Riemann-Hilbert problem. Another approach, due to
Adler, Ferrari, Johansson, van Moerbeke and Vet\H o in a series of papers,
expresses the kernel in terms of resolvents and Fredholm determinants of the
Airy integral operator acting on a semi-infinite interval $[\sigma,\infty)$,
involving some objects introduced by Tracy and Widom. In this paper we prove
the equivalence of both approaches. We also obtain a rank-2 property for the
derivative of the tacnode kernel. Finally, we find a Riemann-Hilbert expression
for the multi-time extended tacnode kernel.
