  Context. Chromospheric fibrils are thin and elongated structures that connect
nearby photospheric magnetic field concentrations of opposite polarities. Aims.
We assess the possibilities and drawbacks related to the use of current
instrumentation and inversion techniques to infer the thermodynamic structure
of chromospheric fibrils. Methods. We employed spectroscopic observations
obtained in the Ca ii 854.2 nm line with the CRISP instrument at the Swedish
1-m Solar Telescope and in coordination with observations in the ultraviolet Mg
ii h & k lines taken with the IRIS satellite. We studied the temperature
sensitivity of these chromospheric lines to properly invert their spectral
profiles with the Stockholm inversion Code and determine the temperature,
line-of-sight velocity, and microturbulent velocity of manually traced
chromospheric fibrils present in the field of view. Results. Fibril-like
structures show a very particular dependence of their temperature as a function
of the position along their length. Their temperatures at the detected
footpoints are, on average, 300 K higher than the temperature at the midpoint.
The temperature variation appears to be almost symmetrical in shape, with
partially traced fibrils showing a similar trend for the temperature variation.
Additionally, the response of the Ca ii 854.2 nm line core to variations of the
temperature for the inverted models of the atmosphere in fibril areas seems to
be insufficient to properly resolve the aforementioned temperature structure.
Only the addition of more temperature sensitive lines such as the Mg ii h & k
lines would make it possible to properly infer the thermodynamic properties of
chromospheric fibrils. Comparisons between the results obtained here and in
previous studies focused on bright Ca ii K fibrils yield great similarities
between these structures in terms of their temperature.
