jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pThe production cross-section of jats:italicJ/ψ</jats:italic> pairs in proton-proton collisions at a centre-of-mass energy of jats:inline-formulajats:alternativesjats:tex-math$$ \sqrt{s} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mis</mml:mi> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula> = 13 TeV is measured using a data sample corresponding to an integrated luminosity of 4jats:italic.</jats:italic>2 fbjats:supjats:italic−</jats:italic>1</jats:sup> collected by the LHCb experiment. The measurement is performed with both jats:italicJ/ψ</jats:italic> mesons in the transverse momentum range 0 jats:italic< p</jats:italic>jats:subT</jats:sub>jats:italic<</jats:italic> 14 GeV/jats:italicc</jats:italic> and rapidity range 2jats:italic.</jats:italic>0 jats:italic< y <</jats:italic> 4jats:italic.</jats:italic>5. The cross-section of this process is measured to be 16jats:italic.</jats:italic>36 ± 0jats:italic.</jats:italic>28 (stat) ± 0jats:italic.</jats:italic>88 (syst) nb. The contributions from single-parton scattering and double-parton scattering are separated based on the dependence of the cross-section on the absolute rapidity difference ∆jats:italicy</jats:italic> between the two jats:italicJ/ψ</jats:italic> mesons. The effective cross-section of double-parton scattering is measured to be jats:italicσ</jats:italic>jats:subeff</jats:sub> = 13jats:italic.</jats:italic>1 jats:italic±</jats:italic> 1jats:italic.</jats:italic>8 (stat) jats:italic±</jats:italic> 2jats:italic.</jats:italic>3 (syst) mb. The distribution of the azimuthal angle jats:italicϕ</jats:italic>jats:subCS</jats:sub> of one of the jats:italicJ/ψ</jats:italic> mesons in the Collins-Soper frame and the jats:italicp</jats:italic>jats:subT</jats:sub>-spectrum of the jats:italicJ/ψ</jats:italic> pairs are also measured for the study of the gluon transverse-momentum dependent distributions inside protons. The extracted values of ⟨cos 2jats:italicϕ</jats:italic>jats:subCS</jats:sub>⟩ and ⟨cos 4jats:italicϕ</jats:italic>jats:subCS</jats:sub>⟩ are consistent with zero, but the presence of azimuthal asymmetry at a few percent level is allowed.</jats:p>